C1 → C1 - C2 (i.e. Replacing 1^st column by subtraction of 1^st and 2^nd column)

C2 → C2 - C3 (i.e. Replacing 2^nd column by subtraction of 2^nd and 3^rd column)

We have a³ - b³ = (a - b)(a² + ab + b²) and b³ - c³ = (b - c)(b² + bc + c²)

Therefore taking (a - b) and (b - c) outside the determinant from 1^st and 2^nd column respectively

C1 → C1 - C2 (i.e. Replacing 1^st column by subtraction of 1^st and 2^nd column)

As a² - c²) = (a + c)(a - c) therefore taking (a - c) outside the determinant from 1^st column we get

.

Expanding the determinant along 1^st row

∴ LHS = (a - b)(b - c)(a - c)( - (a + b + c))

Adjusting the minus sign with (a - c)

∴LHS = (a - b)(b - c)(c - a)(a + b + c) = RHS

∴